Lecture 6: Screening with continuum of types

Lecture 6: Screening with continuum of types

Ram Singh

Department of Economics

February 2, 2015

Questions

We are ready to address the following questions:

Question

How can we solve the problems involving many (possibly infinite) types of agents?

Do the previous results -on rent-extraction, allocative inefficiency, and efficiency-rent trade-off - hold for more complex settings?

Does allocative efficiency and rent extraction increase with the type?

Do contracts always satisfy the monotonicity (of consumption or production)?

When is bunching likely to emerge?

Continuum of Types I

Suppose,

θis drawn from the support[θ, θ]

F(θ)andf(θ)are cdf and density functions, respectively.

Payoff function of an agent (buyer) isu(θ) =θv(q(θ))−T(θ).

The menu offered contracts is(q(θ),T(θ))

From the revelation principle it follows that the principal’s problem is

q(θ),Tmax(θ)

{ Z θ

θ

[T(θ)−cq(θ)]f(θ)dθ}

s.t.

(∀θ,θˆ∈[θ, θ]) [θv[q(θ)]−T(θ)≥θv[q(ˆθ)]−T(ˆθ)] (ICs) (∀θ∈[θ, θ]) [θv[q(θ)]−T(θ)≥0] (IRs)

Continuum of Types II

From the previous analysis we know that(IR)will bind only for the lowest type. It is straight to check thatICimplies that for all

(∀θ > θ)[θv[q(θ)]−T(θ)>0].

Therefore, the aboveIRscan be replaced with

θv[q(θ)]−T(θ)≥0. (IR⁰)

Implementable Allocations: Prosperities I

Definition : An allocation(q(θ),T(θ))is implementable iff it satisfies(IC), i.e., iff

(∀θ,θˆ∈[θ, θ]) [θv[q(θ)]−T(θ)≥θv[q(ˆθ)]−T(ˆθ)].

Proposition

Assuming that U(.)satisfies the single crossing condition, an allocation (q(θ),T(θ))is implementable iff

(∀θ∈[θ, θ]) [θv⁰[q(θ)]dq(θ)

dθ −T⁰(θ) =0] (0.1)

and dq(θ)

dθ ≥0. (0.2)

The proposition says thatICabove is equivalent to (0.1) and (0.2), where (0.1) called the local incentive compatibility constraint and (0.2), of course if the monotonicity condition.

Implementable Allocations: Prosperities II

Proof:Suppose(q(θ),T(θ))is implementable, i.e., (IC) hold. We prove the claim for our specification of the utility function for the agent (the buyer). Let

W(θ,θ) =ˆ θv[q(ˆθ)]−T(ˆθ),i.e.,

W(θ,θ)ˆ denotes the buyer’s payoff when his actual type isθbut he announces/pretends his type to beθ.ˆ

Assuming thatq(θ)andT(θ)are differentiable functions ofθ, andW(.)is differentiable function ofq andT, the buyer’s problem can be looked at as choosingθˆto maximizeW(.). Now,

(q(θ),T(θ))is implementable implies that dW(θ,θ)ˆ

dθˆ |θ=θˆ =dW(θ, θ)

dθˆ =0,i.e.,

Implementable Allocations: Prosperities III

{θv⁰[q(ˆθ)]dq(ˆθ)

dθˆ −T⁰(ˆθ)}θ=θˆ =θv⁰[q(θ)]dq(θ)

dθ −T⁰(θ) =0 This is foc. The soc will gives us

d²W(θ,θ)ˆ

dθˆ² |θ=θˆ ≤0,i.e., {θv⁰⁰[q(ˆθ)](dq(ˆθ)

dθˆ )²+θv⁰[q(ˆθ)](d²q(ˆθ)

dθˆ² )−T⁰⁰(ˆθ)}θ=θˆ ≤0,i.e., θv⁰⁰[q(θ)](dq(θ)

dθ )²+θv⁰[q(θ)](d²q(θ)

dθ² )−T⁰⁰(θ)≤0. (0.3) Differentiating (foc) w.r.tθ, we get

θv⁰⁰[q(θ)](dq(θ)

dθ )²+v⁰[q(θ)]dq(θ)

dθ +θv⁰[q(θ)](d²q(θ)

dθ² )−T⁰⁰(θ) =0 (0.4)

Implementable Allocations: Prosperities IV

From (0.3) and (0.4),

v⁰[q(θ)]dq(θ)

dθ ≥0,i.e.,dq(θ) dθ ≥0, in view of the fact thatv⁰(.)>0. That is, (0.2) holds.

Remark

For our specification of the agent’s payoff function we have shown that (IC) implies (0.1) and (0.2). More generally, (0.1) and (0.2) follow from the single-crossing condition.

Now suppose(0.1)and(0.2)hold.

Assume the contrary to (IC), i.e., assume that

(∃θ,θˆ∈[θ, θ]) [θv[q(θ)]−T(θ)< θv[q(ˆθ)]−T(ˆθ)],i.e.,

∃θ,θˆ∈[θ, θ][W(θ,θ)ˆ >W(θ, θ)].

Implementable Allocations: Prosperities V

Letθ <θ.ˆ

W(θ,θ)ˆ −W(θ, θ) = Z θ^ˆ

θ

∂W(θ,x)

∂x dx=

Z θ^ˆ

θ

θv⁰[q(x)]dq(x)

dx −T⁰(x)

dx

By(0.2), ^dq(x)_dx ≥0, and

x > θ⇒xv⁰[q(x)]> θv⁰[q(x)].

Therefore, we have Z θ^ˆ

θ

θv⁰[q(x)]dq(x)

dx −T⁰(x)

dx<

Z θ^ˆ

θ

xv⁰[q(x)]dq(x)

dx −T⁰(x)

dx and

Z θ^ˆ

θ

xv⁰[q(x)]dq(x)

dx −T⁰(x)

dx=0,

Implementable Allocations: Prosperities VI

since by(0.1),(∀x)h

xv⁰[q(x)]^dq(x)_dx −T⁰(x) =0i

. Therefore, Z θ^ˆ

θ

θv⁰[q(x)]dq(x)

dx −T⁰(x)

dx<0,i.e., (∀θ <θ)[Wˆ (θ,θ)ˆ −W(θ, θ)<0].

Caseθ >θˆis analogous. That is,

θ, θ][W(θ,θ)ˆ <W(θ, θ).

Therefore, we get a contradiction. Q.E.D.

Second Best: Solution I

Coming back to the principal’s problem, in view of Proposition 1, the principal’s problem can be written as

max

q(θ),T(θ)

(Z θ

θ

[T(θ)−cq(θ)]f(θ)dθ )

s.t.

[θv[q(θ)]−T(θ)≥0] (0.5)

dq(θ)

dθ ≥0 (0.6)

(∀θ∈[θ, θ])

θv⁰[q(θ)]dq(θ)

dθ −T⁰(θ) =0

(0.7) Note that at the optimum,(0.5)will bind. Also,(0.6)and(0.7)are equivalent to (IC), i.e.,

θv[q(θ)]−T(θ) =max

θˆ

θv[q(ˆθ)]−T(ˆθ).

Second Best: Solution II

Also note that maxθˆ

θv[q(ˆθ)]−T(ˆθ) =θv[q(θ)]−T(θ) =W(θ, θ) =W(θ).

Therefore, by envelop theorem, dW(θ)

dθ =v[q(θ)] +θv⁰[q(θ)]dq(θ)

dθ −T⁰(θ) In view of(0.7),

dW(θ)

dθ =v[q(θ)].

Now, note that Z θ

θ

v[q(x)]dx = Z θ

θ

dW(x)

dx dx=W(x)|^θ_θ=W(θ)−W(θ).

Second Best: Solution III

Therefore,

W(θ) = Z θ

θ

v[q(x)]dx+W(θ),i.e.,

W(θ) = Z θ

θ

v[q(x)]dx, (0.8)

since(0.5)binds, i.e.,W(θ) =0.

Therefore, fromW(θ) =θv[q(θ)]−T(θ)it follows that T(θ) =θv[q(θ)]−

Z θ

θ

v[q(x)]dx.

Ignoring(0.6)for the time being, the principal’s problem is max

q(θ)

(Z θ

θ

[θv[q(θ)]− Z θ

θ

v[q(x)]dx−cq(θ)]f(θ)dθ )

.

Second Best: Solution IV

Integrating the second term by parts, we get¹ max

q(θ)

(Z θ

θ

([θv[q(θ)]−cq(θ)]f(θ)−v[q(θ)][1−F(θ)])dθ )

. This implies point-wise maximization of the integrand w.r.t. q(θ), the foc for which is

θv⁰[q(θ)] =c+1−F(θ)

f(θ) v⁰[q(θ)]. (0.9)

Remember, the first best requires

θv⁰[q^FB(θ)] =c.

Therefore,

θ=θ⇒q(θ) =q^FB(θ). But,

(∀θ < θ)[q(θ)<q^FB(θ)], i.e., under consumption for every type exceptθ.

Second Best: Solution V

Moreover,ceteris paribus,q^FB(θ)−q^SB(θ)is proportional to 1−F(θ).

The last inference is true for uniform densities. However, may not hold in general.

Exercise Show that

The information rent enjoyed by agent withθdepends onq^SB(θ),θ < θ In general, the information rent enjoyed by agent withθ, depends on q^SB(ˆθ),θ < θ.ˆ

Hint:It immediately follows from one of the expressions above. Find out the expression.

Second Best: Solution VI

Rewriting(0.9), we get

[θ−1−F(θ)

f(θ) ]v⁰[q(θ)] =c. (0.10)

Let

h(θ) = f(θ) 1−F(θ). Definition

h(θ)

is called theHazard Rate.

It is the conditional probability that the consumer’s type belongs to [θ, θ+dθ]given that he belongs to[θ, θ].

Second Best: Solution VII

(0.10)can be written as

g(θ)v⁰[q(θ)] =c, (0.11) where

g(θ) =θ− 1 h(θ). Differentiating(0.11)w.r.t.θwe get

dq

dθ =−g⁰(θ)v⁰[q(θ)]

v⁰⁰[q(θ)]g(θ) Since by assumptionv⁰⁰(.)<0 andg(θ)>0, ifh⁰(θ)≥0 then

dq(θ) dθ ≥0.

Second Best: Solution VIII

However,h⁰(θ)<0 can hold iff(θ)decreases rapidly withθ.

Remark

Non-monotonicity (of consumption or production) will occur when h⁰(θ)≥0 does not hold for a range ofθ∈[θ, θ]; or

θdirectly affects the payoff for the principal.

In such cases, some bunching will emerge.

1We know thatR_θ

θUV⁰=UV|^θ_θ−R_θ

θU⁰V. LetU=R

v[q(x)]dxandv⁰=f(θ). Therefore, Rθ

θ(Rθ

θv[q(x)]dx)f(θ)dθ= [Rθ

θv[q(x)]dxF(θ)]^θ_θ−Rθ

θv[q(θ)]F(θ)dθ=Rθ

θv[q(θ)]dθ−Rθ

θv[q(θ)]F(θ)dθ= R_θ

θv[q(θ)](1−F(θ))dθ, sinceF(θ) =1 andRθ

θv[q(θ)]dθF(θ) =0